The Italian has slumped to 98 in the rankings from a high of 30 in 2015 and arrived in Switzerland without a tour-level win since January.
But it was Giorgi who proved stronger in the end against world number 23 Suarez Navarro, winning 2-6 7-6 (7/5) 6-2 to set up a last-eight meeting with Aliaksandra Sasnovich.
Top seed Barbora Strycova avoided Suarez Navarro's fate, although she had to save two set points in the second set against Carina Witthoeft before winning 6-2 7-6 (8/6).
Czech 17-year-old Marketa Vondrousova reached the first WTA Tour quarter-final of her career with a 6-1 6-3 victory over Germany's Annika Beck.
Read the original article on Eurosport: Giorgi sees off Suarez Navarro at Ladies Open Biel Bienne